Classifying infants in the Strange Situation with three-way mixture method clustering

Br;t;sl, ]oms/ o/ Prrycholo# (1995). 86, 397-418 Printad in Great BN'tain B 1995 The British Psychological Society

Classifying infants in the Strange Situation

with three-way mixture method of clustering

Pieter M. Kroonenberg*

Depa~fmmt of Edmatio~, L i d m Unt,,w~i(y, W~rrntn~or~c~ye,, 52, PO Box 9555, 2300

RE

Laden, The Nathcrlandr

Kaye

E.

Basford

Drpdrtmmt of Ap~rculturr, The Clncuerrity of Quctnsland

Marion

van Dam

Notiot~al C w t of Audit, 'I-Grattenhgge, Thr, Nethrrlmdr

The quality of the anachment relationship between mother and infant is typically determined in the Strange Situation. The assignments of d a n t s to the A (aroidant), B (secure), and C (resistant) atmchment classes are large1.i but not exclusively based on measurements during the reunion episodes. In this paper, the measurements in the reunion episodes are used to derive a clustering of the infants via three-way mixture method of clustering, 1 technique especially designed far clustering threc- way data (here: infants, variables and episodes). The rcsults are compared with the A - R C classification, and the relevance of the outcomes for attachment research arc

discussed. At the same time, the paper aims to demonstrate the use and usefulness of the three-way clustering procedure Tor data from the social and hchavioural sciences.

In developmental psychology there always has been a strong interest in the

attachment relationship between mother and child. One of the main instruments t o

assess the quality of attachment has been the Strange Situation (see Ainsworth,

Blehar, Waters

&

Wall,

1978).

Measurements are made in several episodes of this

laboratory-based procedure, and the final assessment is a classification of each infant

(or better, infant-mother relationship) into one of three categories, usually indicated

with the letter A,

B,

and

C

(see Ainswonh e t

of.,

especially pp. 334-335, 343-362).

The classification is established via detailed scoring rules using the above-mentioned

measurements.

The major evaluation of the classification procedure as outlined in Ainsworth e t

a/.

(1978,

chapter 6) was to perform

a

port

hoe

discriminant analysis using the

classification as the criterion and the measurements as predictors. A disadvantage of

such a procedure is its circularity. First the measurements are used to create the

classifications, which in turn are evaluated by using the measurements. This was

398

Pieter

M.

Kroonenberg,

Koye

E.

Basford and Marion ,Ian Dam

recognized by the originators, but at the time n o clear alternative seemed to be

available. I n the present paper we will attempt t o provide an assessment of the

classification results without using the same data twice by using a clustering

procedure. Clustering procedures attempt t o find groupings of subjects (objects, etc.;

here: infants) on the basis of the measurements available. The resulting grouping (or

statirtical) classification can be compared with the usual c/inica/ or A - B C

classification to assess the latter (see Sawyer, 1966 for a relevant discussion of clinical

versus statistical prediction; the term 'clinical' is seldomly used in this context, but

was inspired by the paper by Richters, Waters

&

Vaughn, 1988). If the groups in the

A-B-C

classification adequately portray the individual differences in the Strange

Situation, then the classes found by the clustering procedure should correspond

reasonably with A-B-C classification, especially if the Ainsworth groups are natural

clusters. If there is little o r no correspondence, the empirically derived clusters will

contain mixtures of the Ainsworth groups (see also Lamb, Thompson, Gardner

&

Charnov, 1985, p. 214).

As we are dealing with three-way data, our prime clustering tool will be the

mixture rnethod

of

cl~stering for three-lug data (Basford

&

McLachlan, 39856;

McLachlan

&

Basford, 1988). As there are to our knowledge no applications of the

three-way mixture method of clustering outside the field of agriculture and biological

sciences (see e.g. Basford, Kroonenberg

&

DeLacy, 1991), we have included an

appendix with some of the technical details, while a more conceptual introduction

including some remarks about interpretation is given in the main body of the paper.

Unique t o the three-way cluster method is that it can handle explicitly data which

arise from the type of three-way designs that form the basis of the present data set.

In particular, during the two so-called remion episodes of the Strange Situation, five

variables measuring the intensity of the infants' behaviours were scored from

videotapes for 326 Dutch infants. Thus, the data set can be seen as a 326 by

5

by 2

three-way data array.

I n this paper we will first provide

an

expos6 of the three-way mixture method

clustering, and add a short discussion of an ordination technique to present some of

the results graphically, i.e. three-way replicated principal component analysis. Then

the substantive background t o the data will be presented. T o appreciate the results

of the three-way method on these data, we will dwell also on the results for two-way

data, especially because several aspects of the mixture method of clustering can

be more easily demonstrated on two-way data. In particular, we start with analysing

the two episodes separately, and only then continue with the data set as a whole. The

results of the last analyses will be compared with the clinical classification t o evaluate

the characteristics of both the clinical and statistical classifications.

Method

There has bcen s long history of 'grouping' approaches when analysing three-may data, probably

stlning with Tucker & Messick (1963) devising a 'points-of-view' approach where the aim was to seek to partition two-way proximity matrices from several sources into relatively homogeneous subgroups. aggregate within, and then run nn analysis for each subgroup. Carroll & Arabic (1983) devised a method

C/assifyin~<

infants in the S/ranxe Situation lui!h three-way m i s t w e method

of

clrrrteritrg

399

methodoloRy called rrmmaes For fitting a hiernrchicd tree structure to obtain a discrete network representation of proximity d:rca. Dc Soete & Carroll (1989) have made further earensinns to this approach using ultramctric trees (for an overview see Arabic, Carroll & DeSarbo, 1987). As far as we

arc aware, the mixture merhod of clustering is the only clustering technique which handles three-way profile datn (i.e. ruhjccts by variables by conditions datn) directly.

The more usual clustering approaches (a, clustering individuals, say) are the hierarchical ones in u-hich by successive fusion thr number of individuals (and subscqoenrlg individuals and clusters) is reduced by one, until one large cluster remains. As there is no explicit model underlying such clustering procedures, it is exucmely diKicult t o evaluntc the optimal number of clusters and the adequacy of the cluster solution for the data. Moreover, once two individuals are allocated to the same cluster in an hiernrchical clustering, they will never be separated again. This is in contrast with the mixture method nf clusrerinrr where for each number of clusters

-

a new solution is sought indcvendent of the solution with fewer or mare clusters.

L'nder the mixture amroach t o clusterine IEveritt. 1980: Wolfe. 1970).

. .

_{-. .}

, . it is assumed rhat the data at hand can be conaidered as a sample from a mixture of several populations in various proportions. Estlmntes of the paramerers of the undedging distributions can be obtained using the likelihood principlr, and the elements can be allocnrcd t o these populations on the basis of their estimated posterior probabilities of group membership. In this utny, individual observations can be partitioned into a

number of discrere, relarivcly homogeneous grnups. However, if the posterior prohahilit" is less than

a specified value the individuals concerned can remain unallncated but with known probabilities of helonging t o the various groups. The three-way mixture mcthod of clustering t o be presented is a direct gencrnlization of the two-way variant developed by Wnlfc (1970). and the two methods are equivalent for a single sample. Even though the esrimarion o< the two-wny case can he solved with the three-way program, special software exists for the two-wny case (see below).

Three-~uay mixtr<re method

of

ciustering

:

Theory

I n this section we will give an ourline of the three-way mixture merhod of clustering. We will skip some details and present a primarily intuitive and simplified introducrion. A more detailed and general exposition is contained in the appendix. The clusrerin~ method is lrased on the assumption rhar each of the 326 f =

_.

1 V l infants belonrrs to one ofpvossihle sroum. but it is unknown t o which

_.

_....

.

- .

one. Therefore. all one can d o is assign i t to that group to which it has the highest prnbabilitg of belonging. In order to bc able ro d o so, it is necessam to establish the characterisrics of each srouD _{. ~ .} , and the ~robahilitv of

each infant to belong t o each Rm;p. Observations are svsilahle on 5 (=pj variables ar 2

i

= r) sepsmrc rimes or episodes. Thus thc data comprise two vectors of multivariate observations on each infant, one

for each episode. For infant, I (, = I , .

.

.

, n ) these will he denoted by

5,

and

*;,,

respectively. The vector of all observations for infant j has 10 elrments. and will be denoted by r,.

I n t us fimr assume that there exisrs only one group, then the data would have come from two multivarintc normal distributions, one for each episode. When there are morc groups. then thc normal distributions of the variables in ench group are allowed tn have different means in each episode. thus

there are p x r x g ( = 5 x 2 r g ) means to be esrimated. Furthermore, it will be assumed that the multivatiarc distribution for each group will not change between episodes, but groups may or may not have ditTerent covariances, i.e the groups may have a conmoil (within-group) covariance matrix, or each group may have an arbilrar~ (within-group) covariance matrix. Thus there is either one covariance matrix o r there a r e g covariance matrices to be estimated. Under the normal mixture model proposed by Basford & McLachlan (19RSb) for three-way data, iris assumed that the rclarive sizes o i t h e ~ g r o u p s C;,,

...,

G,, are given by the n,ixingprgbortionr n,,

..

..n, respectively, which are also unknown. so that they. too, hnvr to br estimated.

400

Piet~r

M .

Kroonenb~rx,

K g e E.

Basford and Marrott van Dam

A tendency has been observed for the derived clusters to be of roughly equal sire when the covariance matrices are specSed to be equal (see e . 5 Gordon, 1981, p. 52). Clearly, if the model is incorrect, for instance, if the parent populations i r e decidely nnn-normal, the merhod might be h r from optimal (see

e.g. Everirt, 198[I, section 5.2). Whethcr in real life this is a problem depends on the aim of the clusrering, i.e. whether one is srcklng for natural clurters, or wants to dissect continuous observations. I n the latter case, the multinormality nssumption seems a reasonable one ra make, in the former case

it is very much an empiricnl mztter whether the method is appropriate.

Testing for the number of componenrsg in n mixture is an important bur very difficult problcm which has not been completely resolved (Mcl..nchlm & Rasford. 1988). An obvious way of approaching the problem is t o use the likelihood ratio stnristic A, as discussed by Wolfe (1970, 1971), ro test for thc smallest value of g compatible wnh the dara. Many authors, including Wolfe, have noted that unfortvnntcly with mixture models, regularity conditions d o not hold for -2lnA to have its usual nsymptoric null distribution of chi squared with degrees of freedom equal to rhe difference in the number of parameters in the two hypotheses.

McLachlan & Basford (19R8) recommended thar in general the outcome of Wolfe's likelihood ratio teat should not be rigidly interprercd. but rather used as a guide to the possible number of underlying groups. They suggested that use also be made of the estimates of posterior probabilities of group membership in the chnice of,+ Thcp can be examined for vnluer oFg near to the value accepted according to the likelihood ratio test, nnd mnv rherefore be of assistance in leading to a final decision as to the number of underlying groups.

When the number of components in the mixture is known (as mny be the case here with2 = 3, i.e. equal to the number of clinial classification groups), Everirt (1981) stated rhat the parameters in the model 'may be esrimatcd hy maximum likelihood methods although problcms may arisr due to singularities in the likelihood function unless some constraint is placed on the variance-covariance matrices, the most nntural being thar these are the same for all components' (p. 171). Note, however. that although it may be natural to assume equal covrriance mntricer, in many applied cases (including here) the data do not really conform with rhat assumption.

A

detailed discussion oCrhc problem is given in Duda & Hart (1973. np.

. .

198--2011.

The mixture m d e l used herc assumes that the measurements taken on infants during the separate episodes are independent of each orher, in that the likelihood was obtained by multiplying togcther the probability density fi~nction for each infant in each episode. Therc is no concern about the inf~nts being

independent of each other, but the same infants were measured during both episodes. Therefore, we arc really assuming independence of mrnsuremenrs on the same infants ovcr time. Independence was a valid assumption fot the agriculrural data to which the model had been previously applied (the same genotypes, but separate plants, were grown in each locntion), but may be open to criticism with social science dara of this type. Treating rhe separate episodes as independent measurements is a compromise which enables some of the structure of the design of rhr experiment to be accommodated, i.e. the same five variables are measured each time.

Note thar the exploratory nature of the clustering methodology is being stressed here. Although we probably have as much experience as anyone in the application of the three-way mixture method of clustering, we do not know about the robustness of the method to the violation of this independence assumption. It is hoped rhat by application to repeared observation dara, such insight will be gained.

Other clustering techniques simply stack the observations for the two episodes together and consider the dnra array to be two-way (infants by variables). I t the outcome of the three-way mixture appnjach is supported by the outcome of the complementary ordination procedure discussed below, then ?he independence assumption cannot be roo rigid. \Y'e are usinfi an rltcrnative methodology on a particular data set to look at a substantive problem, rather than choosing a data set to best demonsrrate the methodology.

Ordination as a con~plmenf

t o

cc(sf~ring

C/a.r.rifyin~

infants in the Strung? Sifuation with three-way n~ixfure

method

of

rlttst~ring

401

infanra are anly described through the clusters, bur nor on an individual level orhcr than iheir ripoitrriwi probabilities r l i beionging to the groups. By using rlrdinarion tect~niqurr in conjuncrion with cluster analysis, n more dcrnilrd analysis of the indivirluals and of the clusters can be obtained. Within social and bchnviournl sciences, principal component analysis is probably rhe most common, ordination techniqur for two-way dnra. Far three-way darn, ruch as we hare here, scecral variants of principal component analysis exisr, ruch as three-mode principal componenr analysis (see e.g. Kroonenberg, 1983), and parallel hctor analysis (see Hnrshman & I.undy, 1984). .As in the present case we only have two episodes, we will use a "cry simple model for the three-way data. sometimes called rrpffrotd (or

auixbhd) prinr;f~tlrnmponmr mod,/. This model is a special case of both three-mode principal component and parallel f%ctor analysis models (see Ten Bcrge, Dr 1.ceuw & I<roonenberg, 1987).

In replicated principal component analysis, the data from two or more occasions or episodes are assumed to h a w rlle same confr~urntinn for rhe vnnables and for the infants for each occasion, buc the rclative size or importance is allowed lo ~ a r \ from cane occasion t o the next. Note ihnr rhis assumption

tits very nicely wirh the assumption of the mixture method of cluster~ng thar per group thr covariance marris is the snme h r rach occasion. The cluster technique allows interaction between rhc variahler and the occasions by modelling n riiffcreni mean Tor each variable in each group at each occasion. In the replicated component model as used here, the overall mezns of the varinbler for each occasion are

modelled separately, and do not form an integrnl part of the three-way model. As we will are rhe ditTerences across occasions k t w e e n the means of the groups ;arr not extremely large, so thar a rwsonable concordance bcmfern the results of the cluster analysis and the component analysis should be porriblc.

Thc replicated principal component model for three-way data can be written as

whcrr c, is rhe a~eight o r rclative contribution for episode

k,

the st, are called component scores and b,. the component loadings, and j is rhc indcx of the infants, / th,r index far rhe variables, and S the number nf compnnmts; ti,, is the crror of approximarton. Gcnenlly, one is only interested in n small number oicornponents, sag two or three. This partially depends on the dimensionalit? of the space in which the clusters can

br

sho*n t o their greatest advantrgr. The components from rhis technique r i l l Ix used t o make a simultaneous plot of the compnncnt scores, the component loadings, and thc clusters.

In thls way, insight can bc acquired about the distinctness and rightness of ihe clusters in relarion ro

rhe variables and the individual infants.

Computer prograrns

'The three-way mixture merhod of clustering is implemented in the computer program called ~ r s c ~ u s . 3 ,

and can bc obtained from the second author (as can the two-way version of the prngram, ~ r u c ~ u s 2 ) .

.\n earlier version of this prqqmm was published as an appendix in McLachlm & Basford (1988). The replicated principnl component analyses have been carried out with a program for three-mode principal componenr analysis ( T U C K I I L S ~ ) developed by the first author (Kroonenberg, 1994; Kroonenberg & 13rouwer. 19')3).

Substantive background

.Straqqe .Siftdotion procedure

402

Piefer

M.

Kroonenherg, Kayc

E.

Basford

atm' Afarion

van Dam

and RZ), are those in which the mother returns after having been away, and the infant war left alone wirh a stranger. On the basis of the infant's behnviour during the procedure, the qunlity of the infantmother anachment relationship is categorized as i n r c c u r c - ~ ~ ~ o i h ~ z (A), r8c"rf (R), or inrtcurt

rrrirtant (C). It should be noted that a fourth catcgory has recrntly been added, the D or disorganized classification (see Main & Solomon, 1990). However, this classification category will nor be crmsidered here as it has not pet been codcd for our d s n .

Three important claims have been made wirh respect to the reliability and validity of the /\-B-C typology, i.e. the clinical classification. First, different patterns of khaviour in the Strange Situation arise from different previous patterns of inbnt-mother interaction. In particular, it is rhe mother's sensitivity to the behaviour of their infants thar leads to secure attachment. Second. infants seen more than once

in the Srrange Situation tend ro behave in the same farhion each time they are measured. I t should be mentioned that attachment classifications remain stable ovcr a period of one ro six months provided family circumstances are stable. Third, individual differences in Strange Situation behaviour predict behavioural differences in other contexts up to several years later, again provided there hns k e n stability in the family circumstances (see among others, Lamb e f a/., 1985; and Vsn Dam, 1993, for further references). The n h v c suggests thar the clinical clnssificntion has both a certain amount of reliability and validity.

According to the clasrificntion instructions (see Ainsworrh e l of.. 1978. pp. 59-63) the scores on five seven-point scales in the tn,o reunion episodes play n crucial role in the clinical classification, i.e. proximity seeking (PS). contact maintaining (CiLL), resistance (RS), avoidance (rZV), and distance interaction (Dl). where high scores on nvnidancc are especially indicative for an A classificatinn, and high scores on resirtnnce for n C classificarion. Therefore, we will also restrict ourselve~ to these variables (see also Lamb d o l . 1985. pp. 209. and Richters ct a/., 1988).

T h e only other clustering of strange siruation measurements known to us (Lamh e l <I., 1985, pp.

214-221) was also restricted ro these same I0 vildahlm, but Lamb ~ 1 0 1 . used hierarchical two-way cluster methods on a sample consisting of Swedish and American infants. Connell (1977) also claims to have carried out a cluster analysis, but closer inspection shows that in fact he used an ordination technique.

Data

A total of 326 infants, o r rather infant -mother pairs, are included in our analyses. They originate from five different studies conducted at the Centre far Child and Family Studies of the Depsrtment of Education, Leiden University. The primary references for these studies, which also contain the detailed information an procedural questions, are tinossrns (1986; see also Van IJrendaarn, Goosscns,

Krooncnberg & Tavecchio, 1985). Goossens & Van IJrendoorn (1990), Hubbard Sr Van IJzendnorn (1991). Lambermon & Van IJzendoorn (1989). and \'an Dam & Van IJzendoorn (1988); a

comprehensive description can Ix found in V m Dam (19113).

As thc data originate from several samples, whicb were collected for different purposes, the reliability information is not entirely complete for all of the scales in all srudies. A report containing all available derails can be obtained from the first author. A typical examplc of the reliability uf the scoring of rhe five varinbles can be found in \'an IJzendoorn r/ 01. (1985, p. 441). 'The inter-coder reliabilities varied from .73-.97, and similar values have been found in the othcr subsamples. With respect to the ,I-B-C classification. a subset of the same study was independently reclnssified %.hi& gave a correct classification rate of .96, and again similar vnlurs were found it? other rubnmples. In as far as a.13

feasible, the classifications were done by other persons than the scorers of rhe five variables. Correlations between the asme rariables across episodes for the whole sample are given in Table 6, and these values

range from .35 (avoidance) to .71 (distance interaction).

Atfachmenf:

A

continuor~r o r discrete constrrrtt?

C/assi/ring it!fanfs

in the Jtran.e

Sitwation with three-wq

mixture metbod o f c/usterin~ 403

be little theoretical reason to expect that individual differences in Strange Situation are discrete rather than continuous. Working with the protocols of 23 infants. Ainsworth r t a / . (1978) developed the clarsificntion system by grouping infants in clusrerr on the basis of perceived behavioural similarities in the srrangc situation. Similarities betu.een the

result in^

(seven) dusters were then used to achieve a funher condensation to three main groups, i.e. rhe A, Hand C classifications. Afcer rhe clnssification of infants. Ainswnrth c t 01. (1978) identified aspects of behaviour that seemed crucial in distinguishing the various groups and subgroups in the classification. Thesc aspects are primarily those which arc included in our analyses. Thus, the classification instrucrions were the result of a purely infi>rmal empirical exercise without strong o prior, theoretical reasons for a typology. As Connell & Goldsmith (1982) noted 'unless typologies ate derived by appropriate empirical means (e.g. cluster analytic techniques). rhey are unlikely to exhibit rhe same predictive capacity and internal structure in subsequent applications' (p. 219). Sevcrel vescarchrrs question whether individual behavioural differences in the Strange Situarion arc adequately represented by the discrete categorization, and the.: consequently recommend the conrideratinn of continuous measures (Cnnnell & Goldsmith, 1982; Kroonenberg &

Van IJrendoorn, 1987; Lamb e t 01.. 1985).

Irrespective of the discrercoess of the atrachmenr construct. clu::ter methods will produce clusters. either by dissecting continuous dimensions o r by seeking for natural clusters. Only afterwards, can

one discern rhe nature of the clusters, for insrance by using ordinarion methods. Results of the cluster analyses therefore will not provide il definite nnswcr to rhe (still unresolved) question whether attachment is a discrete or n continuous construct. I-Iowever, if the groupings derived by the cluster method correspond to the clinical clarsifications, this will at least give funher support to the A - B ~ C typology. If not, we will i r least gain further insight into the individual differences in the Strnngc Situation.

Results

404

Piehr

M .

Kroonenberx,

Kayc E.

Barford and Marion uan Dam

additional information about the basis for the clustering can be gleaned from two-

way discriminant analyses based on all 10 variables.

Anahsis decisions

Number of clusters.

T o demonstrate the considerations that g o into choosing an

optimal number of clusters, consider the first reunion episode under the common

covariance matrix model. One can take some guidance from values of the likelihood

ratio test for comparing a cluster solution with

g

groups with the solution g-1

groups (see McLachlan

&

Basford, 1988, p. 23; \Yiolfe, 1971). Although the log

likelihood increases monotonically with the number of clusters (Table 1). much

smaller gains are made with the addition of more than three clusters (the -21nA

values were 66, 102, and 42 for

g

=

5

t o 7, respectively). In this subsection,

information on five to seven groups is discussed, although not presented.

Table

1.

Results of two-way mixture method of clustering first reunion episode--R1

(common covariance matrix model)

I.og likelihoods

with

hierarchical starts

No. of

Log

likelihood of

groups best

solution

HI

H2

1-1

3

H 4

H5

-21nA

Notcr. Lambda is the rario ford groups 2 n d ~ - 1 groups. The degrees oCfreedom for the approximnte chi-squnred test is twice the difference in number of parameters in the m a models. This is 2*5 for the additional mean vector for the common covariance matrix model and would bc 2*(5 means+p5*6) for the additional mean vector and covariance matrix for the nrhitrnry covariance model. Bold type indicates the best solutions.

MI

to 1-15 are the hierarchical clustering tcchniquer using group average,

median, centroid, flexible sorting, and incremental sum of squares, respectively, as :he classiticarion strategy.

Rasford

&

McLachlan

(198511;

McLachlan

& Basford, 1988, chapter

5 )

also look

at the

oueraN correct aNocation rate and the correct allocation ratc for each cluster,

where

allocation rates are defined as weighted sums of the

aposteriori

allocation probabilities

of entities (here infants) t o clusters. I n the present case they d o not seem

to

be very

informative with respect t o the number of clusters, as the overall values are already

very high for each solution from two to seven clusters (.998, ,979, .974, ,964, ,929,

and ,987, respectively). A further possibility might he t o look at the average absolute

within-group correlations to evaluate how well we have succeeded in creating

homogeneous groups. These values for one t o seven clusters are .41, .19, .21, .14,

.14,

Cla.rs$ying infants in thc Strange Situatiott with three-wq mixture ~nethod of clustering

405

Further information on the stability of the division of the infants into groups can

be gained by cross-tabulation of the various partitionings. T o this end, we show the

cross-tabulation of the three-cluster solution against th? other solutions (Table 2). A

Table

2.

Cross-tabulations of cluster solutions against the three-cluster solution first

reunion episode (common covariance matrix model)

2

1

4

Group

1 2 1 2 3 1 2 3 4 1 140 0 140 0 0 1 3 9 1 0 0

2

109

11 0

109

0 0 84 25 0 3 (1

77

0

0

77

0 1

6 7 0

T

219

77

140

109

77

139

85

31

70

NO@. T = Totals

fair amount of nesting of solutions occurs and this continues even when a larger

number of clusters is examined. Apparently there exist fairly definite and stable

boundaries between the groups. This illustrates one o f the strengths of the mixture

cluster method, i.e. with increasing (decreasing) numbers of groups, solutions are not

necessarily a hierarchy, and

it

is an empirical, rather than a method-dependent, issue

whether nesting takes place.

Assembling all the information on the various cluster solutions and using the

stability argument presented below, it seems that either a two-group or three-group

solution (given common covariance matrices) is optimal for the first reunion episode.

As there are three groups in the clinical typology, it was decided to restrict

subsequent cluster analyses to three groups.

A r b i t r a y versus common muariance matrices.

As explained above, the mixture method is

an iterative procedure which uses maximum likelihood estimation. Because such a

procedure is only assured to converge to a local maximum, one has to use several

different starting allocations to (hopefully) find the global maximum. In the present

case, these were obtained by using the grouping at the appropriate level from each

of several different hierachical clustering methods from the statistical package SAS

CLUSTER (SAS Institute, 1985), in particular, group average (HI), median (HZ),

centroid clustering (H3), flexible sorting with beta equal to -0.25 (H4), and Ward's

method (Ward, 1963) or incremental sums of squares (H5). The starting allocations

for the division of infants into groups presented in Table 1 were obtained without

standardizing the variables.

406

Pieter

M .

Kroon~nherg, Kaye

E .

Barford and Marion r,an Darn

the iterative procedure produces the same solution from different starting allocations.

K

starting allocation obtained via Ward's method very often leads t o a solution with

the highest log likelihood. This is not completely surprising as there exist close links

between Ward's method and the mixture method of clustering (see e.g. Gordon,

1981, p. 50). It seems that for the common covariance case only the two-cluster

solution has good stability (see Table

I).

For this data set, the three-cluster solution with arbitrary covariance matrices is

very stable as all starts converged to the same solution. Even when using the clinical

classification as a startifig allocation, the algorithm converged t o the same maximum.

This indicates that the clinical classification is suboptimal for the mixture modelling

of this data.

At the three-cluster level (as at other levels), the log likelihood of the arbitrary

covariance solution by far outstrips that of the common covariance one (6517 versus

-2743), in other words the assumption of a common covariance matrix is not

appropriate here. Both models produced

a

solution with generally unambiguous

allocations of infants to groups although the partitions were quite different (Table 3).

Table 3.

Cross-tabulations of three-cluster solutions

R

1

R1

R2

Arb~cmry

Common covariances

Arbitrary covariances

covarlances

Group

1

2

3

I

2

3

T

N o t e . T = Totals.

Given the more consistent convergence of the arbitrary covariance matrix model

to a particular solution and the above information o n more definite allocation into

groups,

it

was decided t o continue with the arbitrary covariance model for all other

analyses.

Sepurate analy.res,for the reunion episodes (arbitrary covariance matrices)

Classfiirig infants in

the

Strange Situation with three-11,ay mixture method of clusteriq

407

the clusters are rather diEerent, superficial inspection of the means for each group

indicates that the groups are not unalike (Table 4).

K'e have used the two mixture

solutions from the separate episodes as additional starting allocations for the three-

wag clustering t o see whether they give rise to different solutions in the combined

analysis.

Table

4.

Estimated means for three-cluster solutions for first and second reunion

episodes (arbitrary covariance matrix model)

Group

PS

CM

RS

AV

Dl

N

First reunion episode

1 2.2 1.0 1.7 3.7 4.9 1 85

2 4.0 2.8 1.8 2.8 5.3 45

3 5.6 4.8 2.8 1.7 1.0 06

Second reunion episode

I 1.9 1.0 1.9 3.6 5.3 124

2

4.2 3.6 2.4 2.8 4.5 65

3 5.5 5.4 3.1 1.8 1.0 137

Note. All means greater than 3.5 are indicated in bold. PS = proximity seeking; C M = contact maintaining; R S = resistance; hV = avoidance; Dl = distance intcraction.

First and second reunion episod~s

joint!),

For thc single episode analyses we had n o obvious starting values for group

membership, so had to make do with those from several hierarchical clustering

procedures. For the combined analysis, we can supplement the starting allocations

obtained from hierarchical clustering procedures with the groupings obtained as

solutions from the mixture analysis of the separate rpisodes. As it turned out,

aN

starting allocations (for which the program converged t o a solution) converged to

the same three-way cluster solution with arbitrary covariance matrices. Given the log

likelihood value for one group was

-6041, the best two-cluster solution had a log

Table

5. Cross-tabulations of cluster s o l u t i ~ ~ n s

against joint solution for first

+

second

reunion episodes (arbitrary covariance matrices model)

408

I'ieter

M .

Kroonenberg,

Kaye

E.

Basford and Afarion uan Darn

likelihood of

336, while the log likelihood for the three-cluster solution was 5811.

Solutions with a larger number of clusters only produced marginal increases in the

likelihood. In comparison, the three-cluster solution for the common covariance

matrix model had a likelihood of

-5733. Thus, the three-cluster solution with

arbitrary covariance matrices is clearly the preferred solution. This solurion was also

satisfactory from the point of view of

corrcrf aNocation rater

as these were all equal to

1.00, indicating that all infants were allocated t o groups with an

a posteriori

probability of

1.00.

The cross-tabulation of the R1

+

R 2 three-cluster solution with that of the first and

second reunion episodes (Tahle 5) gave a percentage agreement and G)hen's kappa

of

76 per cent and

K

=

.64 with

R1,

and

82

per cent and

K =

.73 with R2. This

mediocre level of agreement might he disconcerting fnr natural clusters, but not

necessarily so when the clusters are the result of dissections of continuous

dimensions. I n that case, it is easy to imagine that even small shifts in the variables

could lead t o different optimal solutions for the clustering algorithm. 11s the three-

way solution had virtually perfect allocation of the infants t o clusters,

we

are

prepared to put more trust in that solution than in each of the separate ones.

T h e correlation matrices for the R1 and R2 episodes and the three-group mixture

solution are listed (Table

6) to provide an indication of the relationships between

Table

6. Correlation matrices

Or&inal rorrellrtion nlafrices

Re~nion Episode I R~r,niorr Episode

2

Proximity (PS)

1.00 1.00

Contact

(CM) .71 1.00 .71 1.00

Resistance

(RS) .27 .41 1.00 .28 .35 1.00

Avoidance (AV)

-.53 -.49 -.07 1.00 -.59 -.48 .O1 1.00

Distance (Dl)

-.49 -.62 -.32 .19 1.00 -.60

-.69

-.38 .31 1.00

Correlations between the same variables across

episodes

.52 .67 .40 .35 .71

Threr-gro~p solution for R 1

+

R2 orbitray rof~oriot~~e n~africej ntodel

Group 1 Gror~p

2

Proximity

I .OO 1.00

Contact

,013

.no

.51 1.00

Resistance

-

.8

.(I0 1 .OO .10 .13 1.00

.Avoidance

-.23 .OO .33 1.00

-

.41 -.25 .I1 1.00

Distance

.21

.no

- . 1 1 -.45 1.00 -.28 -.44 -.ZR .01 l.n0

Group 3

Proximity

1.00

Contact

.17 1.00

Resistance

.03 .27 1.00

Avoidance

- 3 7 -.I7 .I0 1.00

Classgvit~,q

ir!/ants in

/lie

Strange .Situation with three-way mixture method

of

rltistering 409

variables in the different episodes and clusters. I n both the first and the third group,

all infants had the samc score of 1.00 (the lowest possible) for one of the variables

(contact maintaining and distance intcraction, respectively).

The mean differences of proximity seeking (PS), contact maintaining

(CM), and

distance interaction (DI) contribute most to the distinction between groups (Table

7).

Resistance (RS) and avoidance (AV) are less important, be

it

that they follow the

pattcrn of PS and CM, and that of DI, respectively. These means (Table 7) show that

the

first cluster is characterized by high values in both episodes for avoidance and

distance interaction, and low t o n o proximity, resistance, and contact maintaining.

The

second cluster shows a stable low RS, but increasing PS and CM coupled with

decreasing AV and DI. Finally, the

third cluster has consistently high PS and

ChZ

scores with low to n o AV and

Dl coupled with a comparatively high level of RS.

Overall, there seems t o be a single (proximity +contact) versus (avoidance+distance)

dimension which the cluster method uses to define groups.

Table 7.

Estimated means for three-cluster solution for first+secnnd reunion

episodes (arbitrary covariance matrices model)

Group

PS

CM

RS

AC' Dl N 1 R ' l 2.1 1.0 1.5 3.6 5.1 114 R2 1.9 1.0 2.0 3.6 5.2 2 R l 3.0 1.8 2.0

3.3

4.7 122 R2 4.4 3.9 2.5 2.6 3.3 3 R1 5.5 4.8 2.8 1.8 1.0 90 R2 5.7 5.7 3.3 1.7 1.0

Note. PS = proximity seeking; Cbf = contacr mainnininy; R S = resistance; AV = avoidance;

Dl = distance inrcmction.

Piefer

M.

Kroonenber~, K y e

E.

Basford and Alarion oan

Dam

-0.15

1

I I I I

-0.15 0 . 1 -0.05 0 0.05 0.1

First component

Figure 1. Joint representation of the infants atid variables on the components of the replicared component analysis. Lnhnts are labelled according to their statisucsllg derived cluster membcrship from the ihrce-way mixture method of clustering with the arbitrnrg covariancrs model (@ =cluster 1 : - = cluster 2; = cluster 3; AV = avoidance; CM = contact maintaining; Dl = rlistmce intcr- action: PS = prorimiw seeking: RS = resisrance).

Dl contrast in the cluster analysis. In Fig.

1 the two components are presented, and

each of the infants is labelled accord~ng

to its cluster membership. In the figure, we

have also drawn the vectors of component loadings for the five variables.

Comparison

with

clznical classification

As mentioned in the introduction, the main object of the paper is to evaluate clinical

classification which was constructed using the guidelines set out by Ainsworth e t

a/.

Classijying infants in the Strange Sitnation with three-ivy mixture method

of

rlrcrtering

41

1

Table 8.

Clinical classification versus cluster classitication first and second reunion

episodes combined (arbitrary covariance matrices model)

Group 1 2 3 T A 58 19 3 80 B 66 41 102 209 C 0 5 32 37 T 124 65 137 326 - - Noh. T = Toralr.

T o evaluate why the clustetings are so different, one may look at the means for the

A,

B,

and

C

groups (Tahle 9). The means of avoidance and resistance are notable

when compared with their role in the clusterclassification (Table 7) where they were

less disparate. Further insight can be gained by performing discriminant analyses

on all 10 variables with the clinical classification and the mixture classification as

dependent variables, respectively. The concordance between the mixture cluster

analysis and the discriminant analysis need not be exact. The cluster method treats

thc data as multivariate measurements on the same infants at two separate

(independent) times, whereas the discriminant analysis assumes (a larger set of)

multivariate observation:; o n the infants (at one time) only.

A Further point is that

discriminant analysis assumes equal covariance matrices in the three groups, while we

have shown that the groups found by the cluster analysis have quite different

covariance matrices. I n principle, one would need a quadratic discriminant analysis

t o d o the arbitrary covariance matrices solution justice. However, in the present case

this option is not available as the covariance matrices of both group

1

and

3 are

singular, which prohibits such an analysis.

Table

9.

Estimated means for clinical classification

Group

PS

CM

RS

A V

DI

N

Noto. PS = pcoximi? seeking; CM = contact maintaining; RS = resistance; A V = avoidance;

Dl = distance interaction.

I

_{From the standardized coefficients in the discriminant analyses (Table I@),}

_it

_is

Table 10. Discriminant analyses results (standardized discriminant coefficients)

_{'J fi.} 2

-3

1st

reunion

episode

2nd

reunion

episode

_%

No.

AV RS

CM

P S

Dl

AV RS

CM

P S

Dl

R, %

h

2

Q a rn c/jnicd/ ~Iaf~ification

*

3

All

Vars

I

.4

.1 .O

.2

.0

.8

.2

-.2

-.3

-.2

.78

83.7%

2 A + R / R l 2

1

.3

.I .9

.2

.77

82.2%

@

A + R / R ? 1

1.0

.2

.76

82.5%

8

A + R / R l

2

.9

-.4

.47 66.8%

k

tn

.ill Vars

2

.I .5 -.2

.2

.2

-.2

.7

.1

.0

-.I

.63

ts

A + R / R l Z 2 .I .5

-.2

.7

.62

3

A + R / R 2

2

-.I

1.0

.58

&

A + R / R l 1 .5 .9

.52

9

,I~fi.vture method clurt~ring (arbitray cot'driance motricef mod^()

%

All

1

-.3

.O

.3

.2

-.5

-.I

.I

.3

.3

-.2

.91

88.0%

3

!?.

:\\I

2

.1 .1

-.3

-.l

.6

.2

-.I

.6

.5 .O

.57

2.

3

Aktei.

K,

= canonical cnrrelation; "/o = percentage correct classification by borh discriminant functions jointly; A + R = avoidance md resisrance only:

ti

a R l 2 = R1 + R 2 ; No. = numkr of discriminant function; PS = proximity seeking; CM = contact maintaining; RS = resistance; .4V = avoidance; DI =distance interaction. Values grearer than or equal to .4 in absolute value arc shown in bold face.

9

CLasr{fing i ~ f a n f s

in

the

Jfrange Situation

with

three-wgy mixture

method

of

rlustering 413

are able t o discriminate as well as all variables together, 82.5 and 83.7 per cent,

respectively. The clustering solution, however, is based on the information in all

variables with the least weighting on resistance. Note that while the clustering

method indicates all subjects have aposteriori probabilities of

1.00

of belonging t o

a

articular

cluster, the discriminant analysis casts doubt about the proper allocation

of 12 per cent, i.e.

39

infants. This discrepancy is probably due t o the different

assumptions about the covariance matrices, as mentioned above.

A

final point about the clinical classification can be best illustrated by presenting

again the first two components of the replicated component analysis, but now

labelling the infants in the plot with their clinical classifications. Again the vectors

of the five variables are displayed as well. In the section on the substantive

background we mentioned the problem of continuity versus discreteness of the

attachment construct, which corresponds to the question of natural clusters o r

dissecting continuous dimensions. Inspecting both Fig. 1 and Fig. 2, we see that in

neither case is it easy to maintain that the partitions correspond t o natural clusters.

Both the partitioning by the cluster method and that by the clinical classification

appear to dissect the continuous dimensions.

.

.

. .

_..

.

**

IC t-

-

-

- -

- -

-

.-I--

-

r*

•

E-

--

2

-

--

-.

--

-CM

-

-i- -L

a

-

<-

Dl-

- -

-

I--

- p s

-

_-

=

--

-

a:--

- - -

-

_-

--

-

-

- -

_-

-

-

-

- --

-

-

-

-

-

-

-

- 0

-

-

- -

-

- =

-

- -

=.--

- -- - ---

_-

_{- -}

-

_-

-

-0.15 I 1 1 I 0 . 1 5 -0.1 - 0 0 5 0 0.05 0.1 First component

Figure 2. Joint representation o f the inhnrs and variables on the componenrs of rhe replicated component analysis. Infants are labelled

accord in^

ro their clinical classifications (0 = ,A; - = R ;

W = C; A V = aroidancr; CAI = contact maintaining; Dl = disrance interaction; PS =proximity

414

Pieter

M .

Kronnenber~,

Kqye

E.

Bagfnd and

Alarion won

Darn

Conclusions

Given the results presented, we have t o conclude that three-way clustering does not

correspond t o the

A-R-C

typology (see also Lamb

et a/.,

1985).

The three-way

clustering methods and clinical classification procedures create entirely different

groups of infants. In the course of analysis, these discrepancies were considered from

different perspectives

tr,

gain further insight into the individual differences in the

Strange Situation. From the estimated means for the cluster solutions, it could be

derived that the role of avoidance and resistance is much more important t o the

clinical classifications than to the clusters. The same conclusion can be drawn from

the results of discriminant analyses with the clinical classifications and the clusters as

dependent variables. The two groupings weighted the variables in a different way;

whereas the clinical groupings rest almost exclusively on avoidance and resistance,

the cluster groupings are based on the information of all variables, except resistance.

From the results of a discriminant analysis, which predicted the clusters using the

first principal component of a replicated principal component analysis, one single

dimension appeared t o underlie the clustering. This dimension could be interpreted

as the extent to which infants primarily seek proximity and contact with the mother

(i.e. use PS and Chi), o r primarily stay at a distance from her (i.e. use AV and Dl).

This points towards different styles of behaviour of the infants, largely independent

of their attachment classifications.

The differences between the two classifications point to different underlying

assumptions. Clearly, there are theoretical substantive arguments why resistance and

avoidance play such a dominant role and carry so much weight in the clinical

classification. In the clustering method,

i t

is the differences in the sizes of variances

which determine for a large part the outcome of the analysis, and especially resistance

is a variable with one of the smaller variances. One way of looking at these results

is that there are more and larger differences in the strange situation than are captured

by the clinical classification. O n the other hand, they ate apparently not the ones

which are deemed the most important in the theory of attachment. Earlier, we

indicated that attachment research has shown the validity of the clinical classification

by relating it to several preceding and subsequent behaviours. Whether this can also

be said for the differences highlighted bv the clustering procedure is a matter ro be

investigated.

With respect t o the continuity-discreteness argument, the analyses lend some

support t o the statement by Lamb et a/. (1985), that the A, B, and

C d o not represent

distinct types of infants, but that they have arisen from an

undetlying continuum

which has been artificially trichotomized. Of course, a similar statement can be made

about the statistically derived grouping. These results d o not necessarily imply that

a natural trichotomy does not exist, they only indicate that such a division is not

strongly supported by the present empirical investigation. One should either call

upon theoretical arguments o r additional empirical information to substantiate the

natural clusters claim.

C/a.rsifying infants in the .Ttrun,e .Tituation with three-way mixfirre ntethod

of

r/usteritrf: 41

5

does not appear to he too restrictive, given the consisrencg of the results from

different analytical techniques. Thus, for the present data, we have reasonable

confidence m the summarization in terms of relatively homogeneous clusters.

Acknowledgements

T h c data set, and all its suhsamples, used in this paper originate from the Cenrre of Family and Child Studies of the Dep~rtmcnr of Education, Leidcn University. W c are graceful t o the Director of thc Centre, Prof. M. H, van IJzendoom, for making the data set nvnilahlr to us.

The work of the first author was partially supported by grants of the Netherlands Organization of

Scienrific Research (NWO), the Royal Nerherlands .\cadem of Arts and Sciences (KN:\\V), and the i\cademy far the Social Sciences in Australin.

References

hinsworth, iul. D. S. (1990). Some considerntinns regarding theory and assessment relevant t o attachments beyond infancy. In M. T. Greenberg, D. Cicheni & E. >I. Cummings (Eds). R//orhrnen/ in thc Prr,irhaol ).cur: Theory, Kereord,, nnd lt,/en,m/ian, pp. 463-488. Chicago. IL: C'nivrrsity of Chicago Prers.

Ainswarth, M. D.

S.,

Blchar, hl. C., Waters, E. & Wall, S. (1978). P , z / / a r n ~ ~ i l / t a t h m t n t . -4 PIycholoqirnI .S/u<q of //re .S/mngc J'ituotion. I-iillsdale, N J : Erlbaum.

Anderson, T. W. (1984). An Introdurlion fo il.lnlmit:oriote Stoti~tics, 2nd ed. New York: Wileg.

Arnbie, P., Carroll, J . D. & DeSarbu, W. S. (1987). Thrcr-,u"y .Sralingnnd Clnrteriq Beverly Hills, LA:

Sage.

Basford, ti. E., Icroonenberg, P. M. & DeLacy. I. H. (1991). Three-way methods for multinttrihute genotype x environment data: An illustrated partial survey. Field Crops Rescmrh. 27. 131-157. Basford,

K.

E. & hlclachlan, G. J. (1985s). Estimation of nllocatir,n rarer in n cluster context. Journal

?/'the Amwiran .S/~/ir/icnl . 4 r i o ~ / i o r , 80, 28(r293.

Bnsford, li. E. & McLachlan. G. J. (19R5h). The mixture method of clustering applied m three-way data. journnl q( Clrl~~$ra/ian, 2, 109-125.

Basinrd, ti. E. & McLachlan, G. J . (1985~). Likelihood estimation with normal mixture merhads. .dpplicd I'/n/i.~tirr, 34, 282-289.

Carroll,

1.

D. & Arabic, P. (15183). INDCLllS: An individual ditTercnces generalization of the ~ n c 1 . o ~

model and the MAPCLKS algnrithm. P~yrhornctrikn, 48, 157-169. (Reprinted in H. ti. Law. C. X'. Snyder Jr, J. A. Hanie & R. P. McDonald. Eds. Re~earch Alethodi .for ,bf~Itirnode Doto .ilr,o!yrit, pp. 518-534. New York: Prteger.)

Carroll, J. D., Clark, L. 11. & DeSarbn. W'. S. (1984). The represerttarion of three-way proximiry data by singlc and multiple tree structure models. /ourmi oj' Clurrz$cerion. 1 , 25-74.

Cohen, J. (1960). h coeficient of agreement fnr nominal scales. E,l~,~lm;l/ionolord P~ytholoD~iral hlroruroncv/, 20, 37-46.

Connell. D. B. (1977). Individual diffcrcnces in attachment lehaviour: Imng-term srahility and relationships t o language development. Unpublished doctoral thesis. Syracuse. NY: Syracuse

U ~ v c r s i t y .

Connell, J. P. & Goldsmith,

H.

H. (1982). A structural modeling approach to the study of attachment snd Strange Sjtuntion behnviors. In R . N. Emde 8: R. J. Harmon (Eds), ThtDevr/opmcrrt~/Attachr,,mr und .-lfi/ii/it,# %/ern$, pp. 213-243. New 'iork: Plenum.

Dr Soerc. G. & Carroll. J. D. (1989). Illtrametric tree representations of three-was three-mode data. I n

R. G>ppi & S. Holnsco (Eds), ~\f,,ltiuu~y Dnto Arnllisir, pp. 41-26, Amsterdam: North Holland.

Dempsrer. A. P.. Lnird,

N.

M.

& Ruhin,

D.

D. (197:). Maximum likelihood estimation from incomplete

416

Pieter

A I .

Kroonenheq,

Kcye

E.

Rasford

and

Marion

rwn

Dam

Everirr. R. S. (1981). A Monte Carlo investigation of rhe likclihnod rario rcsr For thc number of components in a mixture of normal rlistrihutinns. Mnltivorio/~ Drhziornl Rrrr#rrl,. 16, 171-180. Gordon, A. D. (1981). Cl,~iri/i<~lion. London: Chnpmnn & I-lnll.

Gnossens. F. A. (1986). ?%ir Qrcnl,ty of .'lrrnchr,,,rwt Rcln/iourh;pf

nf

T ~ o r ~ ~ t o r - n l ~ l ~ o/ if;hrkir<< and Nor,- working h&lhcrr and Vomr Aiinri~rt~d I'nr/arr. 1.euvr.n: Accn.

Goossens, F. A. & \'an IJrendoorn, M. H. (1990). Quality of infanrs' attachments to pn,fessionill carcpivrrs: Relation to infant-parent nrraclrment and dny-care char;~crcrirtics. Chi/dDer~e/opn,ef. 61, 832 837.

Flarshman, R. A. & Lundy. M. E. (1984). The PhnAFAc model f o r threc-way factor analysis and multidimensional scaling. In H. G . L a w . C. W. Snyder I r , J . A. Hattic & R. P. McDonald (Eds). Rc.mrrrb ~Clabodi/or ,llultimode Doto A-lnulyiii, pp. 122-284. New l'ork: Pneger.

Ilathau,ay, R. J . (1985), A constrained f<,rmularion of maximum-likelihood estimation for normal mixture distributions. :lnnnlr qf . S l o / i ~ / i ~ ~ , 13, 7'15-800,

Fluhiurd, I:. 0. A. & Van IJzendrmrn, hl. M. (1991). Mnrcrnsl unresponsiveoess and infant cryin%

across the tint nine monrhs: A naturalistic ionpirudinsl stud". I,,/o,/ Uel,nrr;or ~ n d Drr~elopmmf. 14. 299-312.

Kieier,

1.

& \Volfr,wirz, J . (1956). Consirtrncy of the maximum likelihood estimates in the presence of infinitely many incidentill pnrameters. .'ln,rolr ~f,2.l~/brnv,~timl S'/~r/is/;rr. 27. 887-906.

Kiefer. N. M. (1978). Discrete parameter variation: lficient cstimntion of a swirching regression model. Emnomctriro. 46, 427-434.

Kroonenberg, P. M. (1983). Tbnp-modr Principnl Componenr .4rro/yri.r: Thear)l and Applicolionr. Leiden. The Nctherliinds: DS\VO I'rcss.

Krnunenberg, P.M. (1994). T h e T o c a . ~ ~ s line: A suite of programs for three-way analysis. Lbm,nnta/ion,,l .T/n/i.ctits uond Dats .Ina!~irii. 18, 73-96.

Kroonmberg, P. M. & Brouwer, P. (1'193). ~ P _ I Y P . . I C E Unr:r ?Mdn#nI; A P n ~ b ~ e of Threc-yy Prqgromr,

vol. 3: T I I C K A L S ~ hfnnual. Department o f Education, Leiden Llniversity.

Kmonenberg. P. M. & Vnn IJzendoorn, M. H. (1987). Ikploring children's behavior in the Strmge Siruarion. In L.

W'.

C. Tavrcchio &

M.

M. van IJzendoorn (lids), ..lrmcbmenf in .Tariol hrem~orkr: Canhilil~lionr in / h ~ Rawl~~..linrwior~h il//azhmm/ Throcy, pp. 379425. Amsterdam: Nonh Holland. Lamb, hl. I., 'rhompnon. R . A,, Gardner, \Y'. 8; Chnrnov, E. L. (1985). I~jon~-A~olber Aftnclinrmt: The

Or&i and L)ri,rlopnttnlal .I'(qn,/ir~~nrt sjlndi,lidr,nl D~fferencrr in .Ilronfl Silnlrlion Brhovior. Hillsdale, N J :

Erlhaum.

Lilmbcrmon, M . UT. E. & Van IJzendoorn, M. H. (1989). Influencing mother-infant interaction thmuph videotaped or wrirrm instruction: Evaluation of n parent education progmm. Early Rvnnrrh

(%ildhood_Ounr/trly, 4, 4494%.

Main, M. & Solomon. J . (1090). Procedures for identifying infants as dirorganised/disoricntcd during the Ainswnrth Srranpc Situstion. In

M.

T , Greenberg, D. Cicctti & E.

M.

Cummings (Eds), .,lftacbnrrr,/ it, lbc Preiclrool 1'~dr.r: Th~nnr, Ra~nrrh, on0 lnfrrrrret~fion. Chicago: Universiry of Chicago Pvess.

McLachlnn, G. J. & Rasford. K. E . (1908). .\.fixrur< Mc//mdr: Infrrcncr and,4pp/icdionr lo Clnrfrriw,?. New

Yark: Marcel D e k k r r .

Richtcrs, J. E., Waters, E. & Vaughn, R. E. (1988). Empirical classificntion of infant-mother relaticrnships From inreracrivc behnvior and crying during reunion. Child D~swlopmmf, 59, 512-522. SAS lnsrirurc (1085). .SA.Y Urrr'r G,,;dc: S/d/i~/icr, 1 irrioiion S Edilion. Cnry, NC: SAS Institute. Sawyer. J . (1966). Measurement nndpredictiun, clinical nndsmrisrical. Pr/iliolo~icnIRnllr/i~,. 66. 178-200.

Ten Berzc,

1.

hl. F., l>e laeuw. J. & Kroonenherg. P. h.1. (1987). Some new results on principal component analysis of rhrcr-modc data hy means of alternating leasr squares algorithms. P~ythomrtri&, 52. 183- 191.

Tucker, 1,. R. & hIessick. S. (1963). An individual differences model for multidimensiond scaling. P j r c h o n r p t h . 28. 33-367.

V a n Dam, M. (1993). Sccundsire anillyse van de Strange Situation (Secondary analyses of the S t r n n ~ e Situation). Doctoral rhesis. Iriden, The Netherlands: Department of Education, Lcidcn Ilniversity.